Dezhen Song, Ken Goldberg UC Berkeley, United States Anatoly Pashkevich

1. ShareCam Part II: Approximate and Distributed Algorithms for a Collaboratively Controlled Robotic Webcam Dezhen Song, Ken Goldberg UC Berkeley, United States Anatoly Pashkevich State University of Informatics and Radioelectronics, Belarus Supported in part by the National Science Foundation

2. Robot System Taxonomy (Tanie, Matsuhira, Chong 00) Single Operator, Single Robot (SOSR): Single Operator, Multiple Robot (SOMR): Multiple Operator, Multiple Robot (MOMR): Multiple Operator, Single Robot (MOSR):

4. Contents • Related work • Problem definition • Algorithm • Approximation bound • Distributed algorithm • Results • Future work

5. Related Work • Facilities Location Problems • Megiddo and Supowit [84] • Eppstein [97] • Halperin et al. [02] • Rectangle Fitting • Grossi and Italiano [99,00] • Agarwal and Erickson [99] • Mount et al [96] • Kapelio et al [95]

6. Related Work • Similarity Measures • Kavraki [98] • Broder et al [98, 00] • Veltkamp and Hagedoorn [00] • Distributed robot algorithms • Sagawa et al [01], Safaric[01] • Parker[02], Bulter et al. [01] • Mumolo et al [00], Hayes et al [01] • Agassounon et al [01], Chen [99]

7. Related Work • Existing algorithms for ShareCam • Song, van der Stappen, Goldberg [02] O(n2) • Har-Peled, Koltun, Song, Goldberg [03] O(n log n)

8. OneOptimal Frame find optimal frame ShareCam Problem: Given n requests,

9. 3z (x, y) Problem Definition • Assumptions • Camera has fixed aspect ratio: 4 x 3 • Candidate frame c = [x, y, z] t • (x, y)  R2(continuous set) • z  Z (continuous set) 4z

10. Problem Definition Requested frames: ri=[xi, yi, zi], i=1,…,n

11. Problem Definition • “Satisfaction” for user i: 0  Si  1  = c  ri c = ri Si = 0 Si = 1

12. Requested frame ri Area= ai Candidate frame c Area = a pi Satisfaction Metrics • Measure user i’s satisfaction:

13. Optimization Problem

14. Algorithm Overview • Grid based approach • Derive approximation bound • Price to pay for enlarging a candidate frame • Optimal frame must be enclosed by a large frame on the sampling lattice. The size difference depends on lattice resolution • Bound depends on inputs and lattice resolution • Distributed algorithm

15. y x Approximation Algorithm Compute S(x,y) at lattice of sample points: d w, h : width and height, g: Resolution range

16. Approximation Bound Requested frames

17. Approximation Bound c Candidate frame Requested frames

18. Approximation Bound ca cb Candidate frames Requested frames

19. Approximation Bound ca cb Candidate frames Requested regions

20. Approximation Algorithm ca cb

21. Approximation Algorithm c* : Optimal frame : Smallest frame at lattice that enclosesc* • Run Time: • O(n / 3) : Optimal at lattice (Algorithm output)

22. Distributed Algorithms • Server O(n+1/3) • Client O(1/3) • Robustness to dropouts…

23. Distributed Lattice • Define Final Lattice (Define d) d d

24. Distributed Lattice • Divide Lattice point based on n (Assume n=4)

25. Distributed Lattice • Sub lattice for each user

26. Robustness to client failures

27. Results • A demo with 6 inputs t

28. Current & Future Work - Satellite Application

29. Current & future work - Functional Box Sums • Efficient reporting of [Zhang et al 2002]

30. www.co-opticon.net